Arithmetic Volumes of Shimura Varieties

Boston College, Chestnut Hill MA

Investigators

Abstract

The Principal Investigator will study aspects of the theory of Shimura varieties. These are particular kinds of higher-dimensional surfaces whose rich geometry and arithmetic puts them among the central objects of study in modern mathematics. The goal is to find new relations between geometrically defined quantities such as volumes and intersection multiplicities, and quantities that come from other branches of mathematics, like representation theory, that have no a priori geometric meaning. In addition to their importance in pure mathematics, Shimura varieties play an essential role in the Langlands program, which is of increasing interest in theoretical physics, and are important tools for understanding elliptic curves and abelian varieties, which now play a major role in cryptography. More broadly, the project concerns the theory of numbers and arithmetic geometry. This is an area of research which has applications to cyber security, through cryptography, and to some aspects of coding theory. The primary goal of the Principal Investigator's research on Shimura varieties is to prove new relations between their geometric invariants and L-functions. For example, the Principal Investigator will use recent advances in the theory of Borcherds products and integral models to express the arithmetic volumes of Shimura varieties of unitary and orthogonal type as special values of L-functions. The Principal Investigator will also prove new examples of generalized Gross-Zagier formulas, for example by expressing the intersection multiplicities of Shimura curves embedded into the Siegel threefold in terms of the central derivative of a Langlands L-function attached to a cuspidal representation of GSp(4). Similar higher dimensional formulas are also expected. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.